I’ve been thinking about how to engage students in as many different senses as possible when learning math, so as to ensure that students have many distinct memories about a particular topic. The hope is that these different ways of knowing mathematics will create a more robust schema for the information that is acquired, which will then lead to better retention and deeper comprehension.
Ruling out the sense of smell (maybe “scratch and sniff” math books?) and taste, the remaining three senses that can be used to experience math are sight, sound and touch. Students hear math and see math frequently–that’s easy to do. Helping students experience mathematics using their hands or bodies usually requires a bit more creativity.
As I wrote in a previous post, I used Algebra Tiles and the “general area model” to help students calculate the product of two binomials using their visual and tactile senses. But since this approach can be problematic when working with negative quantities or higher powers of variables, it’s best to transition students to more general ways of knowing mathematics as they develop greater skill and understanding.
The ultimate goal is to get students to think of expanding products of polynomials as a generalization of the simple distributive property: a(b+c) = ab + ac. (See note below about “FOIL.”) But how to get students to experience this visually or kinesthetically?
Here’s a silly thing that I did in my Algebra 1 class today: After showing students the “general area model” for expanding products of binomials, I asked two pairs of students to come up to the front of the room. The students’ names start with the letters A, S, E, and G. I purposely chose these students because A and S are close friends, as are E and G, but the two pairs don’t seem to associate with each other much. I wrote on the board (a + s)(e +g), then asked the four students to help me act out a short scene.
The scene: A and S, who are close friends, are going to a party hosted by E and G, who are also close friends. A and S walk in the door and are greeted by E and G. All four people shake hands and introduce themselves and each other.
I then let the students act out the scene in front of the whole class. Some students were shy, which made the whole thing a bit awkward, but they managed to do it just fine. I then asked the class what they saw–in particular, who shook hands with whom?
The students easily pointed out that A shook hands with E and G, then S shook hands with E and G. They also could explain to me that the reason A and S don’t shake hands is that they know each other already (which is the same reason why E and G don’t shake hands).
The point of all of this was to help students kinesthetically experience this product.
Each of the four terms in the answer matches up with one of four handshakes. The diagram (with colored arrows and terms) then became the visual reinforcement for the kinesthetic experience. Maybe a bit hokey, but I think this short demonstration made the idea of using the distributive law very concrete for some students.
Teachers: I’d love to hear what other activities you use to help students engage in mathematics using other senses. One of my close friends gets students to walk and run while graphing their motion, to help students learn the concept of slope.
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Footnote about “FOIL”: The “FOIL” acronym is often taught to students to help them remember how to expand products of two binomials.
(x+2)(x+3) = FIRST (x^2) + OUTER (3x) + INNER (2x) + LAST (6)
It’s a handy way to remember things, for sure, but I am consciously trying not to teach this to my students because it only applies to the product of two binomials–it doesn’t help if you are multiplying a polynomials with more than two terms, for example, (x+2)(x+y+3). It will be interesting to see if students can more easily make the jump to expanding (x+2)(x+y+3) because I’ve consciously avoided teaching “FOIL” and instead focused on this idea of generalizing the distributive property.